A Nonlinear Mechanism for Transient Anomalous Diffusion

TL;DR

This paper introduces a nonlinear diffusion model with concentration-dependent diffusion coefficient, demonstrating that it naturally exhibits transient anomalous diffusion with a crossover from subdiffusive to normal behavior, offering an intuitive local interaction mechanism.

Contribution

It presents a physically-grounded nonlinear diffusion equation that explains transient anomalous diffusion without non-local or fractional models.

Findings

Transient anomalous diffusion observed in the model

Crossover from subdiffusive to Fickian behavior

Mechanism based on local interactions

Abstract

Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local models. This paper investigates a nonlinear diffusion equation where the diffusion coefficient is linearly dependent on concentration. We demonstrate through a perturbative analysis that this physically-grounded model exhibits transient anomalous diffusion. The system displays a clear crossover from an initial subdiffusive regime to standard Fickian behavior at long times. This result establishes an important mechanism for trasient anomalous diffusion that arises purely from local interactions, providing an intuitive alternative to models based on fractional calculus or non-local memory effects.